1The quality of age reporting can be measured by means of age heaping indices to detect the degree of preference or avoidance for certain ages. Among standard indices (Bachi’s, Myers’, Whipple’s, Zelnik’s) [1], Whipple’s index is the simplest to calculate and the most widely applied. It is a summary measure of age heaping on ages ending in 0 or 5 used to determine variability in the quality of age reporting between regions or countries and its evolution over time.

From the original Whipple’s index to a modified version

2The original Whipple’s index is obtained by summing the number of persons in the age range 23 and 62 inclusive, and calculating the ratio of reported ages ending in 0 or 5 to one-fifth of the total sample. It assumes a linear distribution of ages in each five-year age range, i.e. a continuous and linear decrease in the number of persons of each age within the age range considered. Low ages (0-22 years) and high ages (63 years and above), for which the linearity assumption is not plausible, are excluded from the calculation. Whipple introduced the following formula :

3where P_x is the population of age x in completed years.

4If the linearity assumption is satisfied, and in the absence of age heaping, we obtain :

5and

6where ₅P_x is the population of the age range (x, x + 4).

7Assuming that the linearity assumption is verified, W is equal to 1 if no age heaping is observed for ages ending in 0 or 5.

8When there is avoidance of these terminal digits, then P₂₅ < ₅P₂₃/5, P₃₀ < ₅P₂₈/5, etc., the numerator (N) becomes smaller than the denominator (D) and W is less than 1. When avoidance is absolute (theoretical situation in which no reported age ends in 0 or 1), P₂₅ = P₃₀ = … P₆₀ = 0 and W = 0.

9When there is a preference for these terminal digits, then P₂₅ > ₅P₂₃/5, P₃₀ > ₅P₂₈/5, etc., N becomes larger than D and W is above 1. When all reported ages are rounded (another fictitious situation), then N = D and W = 5.

10In this original version, Whipple’s index only measures preferences for (or avoidance of) ages ending in either 0 or 5 without distinction. As these effects may be contradictory and cancel each other out, two changes were made to the formula.

11The first change made it possible to distinguish between preferences for ages ending on 0 and those ending in 5 as follows (Roger et al., 1981, p. 148):

12By summing W_o and W₅ and dividing the sum by 2, we return to the original Whipple’s index (W),

13Though this first modification provides a means to distinguish between preferences for ages ending in 0 and those ending in 5, it is nonetheless based on the unrealistic assumption of linearity over a ten-year age range.

14The second modification, proposed by Noumbissi (1992), returns to a more reasonable assumption of linearity over an age range of five years rather than ten. It is based once more on the underlying principles and assumptions of the original Whipple’s index and introduces the following formulae to measure age heaping :

15Age heaping can thus be calculated for all ten digits (0-9). For each digit, the degree of preference or avoidance can be determined as follows :

16where P_x is the population of completed age x and ₅P_x the population of the age range (x, x + 4).

17If there is no digit preference or avoidance, this “digit-specific modified Whipple’s index” is equal to 1. An index above or below 1 signifies, respectively, preference for or avoidance of the digit in question.

Total modified Whipple’s index (W_tot): an overall summary index

18Though the extension proposed by Noumbissi (1992) made up for the “shortcomings” of the original Whipple’s index by extending its basic principle to all ten digits, it is not practical for spatial, temporal and/or spatio-temporal comparison. We still need a summary index to establish the overall spatial and temporal variability of age reporting.

19Using the digit-specific modified Whipple’s index, denoted W_i, an overall index summarizing all age preference and avoidance effects can be constructed by taking the sum of the absolute differences between W_i and 1 (counting all differences as positive). The total modified Whipple’s index (W_tot) is written as follows :

20where W_i is the digit-specific modified Whipple’s index for each of the ten digits (0-9) developed by Noumbissi.

21If no preference is observed, then

22If all reported ages end in 0 or 5, then W₀ = W₅ = 5 and all other W_i = 0. Hence, W_tot reaches the maximum value of 16 :

23This index can thus be used as a general measure of the quality of age reporting, in complement to Noumbissi’s previous development (1992).

Application to data of variable quality : India, Morocco and Switzerland

24To test this total index, W_tot, we applied it to historical and contemporary age data from censuses in India, Morocco and Switzerland. These three countries present large differences in the quality of age reporting and were chosen for this reason. The quality of age reporting is considered to be poor in India, good in Switzerland and intermediate in Morocco.

25Figure 1 and Appendix Tables A1 and A2 give the total digit-specific modified Whipple’s indices for each sex in the three countries at different dates. Preferences for each digit are illustrated by the digit-specific modified Whipple’s indices (W_i). The value W_tot, in brackets in the legend of Figure 1, summarizes the overall reporting quality for all ages. This summary indicator of overall age reporting quality is a highly valuable tool both for comparing censuses conducted in a single country and for assessing the accuracy of census data from different countries.

Figure 1

Quality of age reporting by sex : digit-specific modified Whipple’s indices (W_i ) and total modified Whipple’s indices (W_tot) in India, Morocco and Switzerland, selected years

Quality of age reporting by sex : digit-specific modified Whipple’s indices (W_i ) and total modified Whipple’s indices (W_tot) in India, Morocco and Switzerland, selected years

26The three countries have different characteristics. Age reporting in India follows a classic pattern, and the strong preference for ages ending in 0 and 5 is reflected in high W_tot indices. The situation in Morocco is less extreme. Preference for 0 and 5 declined over time and had practically disappeared by 1994, as illustrated by the progressive decrease in W_tot. Last, with W_i and W_tot close to 1 and 0, age reporting in Switzerland became very accurate from the second half of the nineteenth century. Slight heaping on 0 and 5 is observed in 1860, but it disappears rapidly. The remaining minor variations can be attributed to fluctuations in population size at different ages that reflect changing fertility, mortality and migration patterns. W_tot thus indicates changes over time in the quality of age reporting within a country. Coupled with the W_i indices, it provides a simple means to identify the particular digits whose changing patterns of preference lead to improved overall age reporting quality.

Figure 2

Total modified Whipple index (W_tot) in India, Morocco and Switzerland, by sex, 1970-2005

Total modified Whipple index (W_tot) in India, Morocco and Switzerland, by sex, 1970-2005

27The index W_tot can also be used for comparisons between countries. Among the three selected countries, we note that, for both men and women, the values for Switzerland are systematically close to zero, while India and Morocco have much higher total values (India has by far the highest values, with the smallest differences between sexes). However, the change in W_tot values over time shows that the general quality of age reporting has improved from once census to the next.

Comparison with Myers’ blended method and the original Whipple’s index

28To assess the total modified Whipple’s index, we will compare it with the other summary measures of age reporting quality. The three following measures are compared : Myers’ blended index, the original Whipple’s index and the total modified Whipple’s index (W_tot) [2].

29Myers’ method also measures preferences for each of the ten possible digits and proposes a blended index. It is based on the principle that in the absence of age heaping, the aggregate population of each age ending in one of the digits 0 to 9 should represent 10% of the total population. The index is calculated by summing the number of people whose age ends with a particular digit for the population aged 10 and over, and then for the population aged 20 and over. Each series is then weighted and the results are added to obtain a blended population. Myers’ blended index is obtained [3] by summing the absolute deviations between the aggregate and theoretical distributions (10%).

30Myers’ blended index is calculated using the same age range (23-62 years) as Whipple’s index (original and W_tot). Last, as the quality of age reporting in Switzerland was high from the mid-nineteenth century, the method was applied solely to Indian and Moroccan census data.

31Though based on different methods, the three indices reveal identical patterns in the quality of age reporting. For India (Table 1) the quality of age reporting falls slightly between 1971 and 1981, then improves again up to 2001, notably in the 1990s. For Morocco (Table 2), quality improves steadily from 1971.

Table 1

Comparison between Myers’ blended index, the original Whipple’s index and the total modified Whipple’s index (W_tot), India, 1971-2001

Myers’ blended index Original Whipple’s index Total modified Whipple’s index (Wtot) Males Females Males Females Males Females Index values 1971 41.45 42.73 2.94 3.00 7.81 8.08 1981 43.90 43.94 3.04 3.05 8.22 8.27 1991 41.50 41.65 2.93 2.88 7.75 7.71 2001 31.64 29.47 2.41 2.18 5.71 5.31 Variation with respect to previous year (%) 1971 – – – – – – 1981 5.91 2.84 3.39 1.56 5.17 2.41 1991 – 5.46 – 5.21 – 3.78 –5.62 – 5.61 – 6.76 2001 – 23.76 – 29.24 – 17.61 –24.13 – 26.33 – 31.12 Male/female ratio of values 1971 0.970 0.979 0.967 1981 0.999 0.997 0.993 1991 0.996 1.016 1.005 2001 1.074 1.104 1.075 Note : The three indices are calculated over the same age range of 23-62 years. Source : Registrar General & Census Commissioner, India (1976, 1987, 1997, 2005).

Comparison between Myers’ blended index, the original Whipple’s index and the total modified Whipple’s index (W_tot), India, 1971-2001

Table 2

Comparison between Myers’ blended index, the original Whipple’s index and the total modified Whipple’s index (W_tot), Morocco, 1971-1994

Myers’ blended index Original Whipple’s index Total modified Whipple’s index (Wtot) Males Females Males Females Males Females Index values 1971 23.88 33.71 2.12 2.58 4.52 6.33 1982 14.32 21.15 1.46 1.79 2.34 3.58 1994 6.18 8.94 1.10 1.29 0.90 1.60 Variation with respect to previous year (%) 1971 – – – – – – 1982 – 40.02 – 37.26 – 31.28 – 30.76 – 48.12 – 43.54 1994 – 56.83 – 57.73 – 24.21 – 27.79 – 61.73 – 55.35 Male/female ratio of values 1971 0.708 0.821 0.713 1982 0.677 0.815 0.655 1994 0.692 0.855 0.562 Note : The three indices are calculated over the same age range of 23-62 years. Source : censuses, data kindly provided by A. Ajbilou and D. Tabutin.

Comparison between Myers’ blended index, the original Whipple’s index and the total modified Whipple’s index (W_tot), Morocco, 1971-1994

32Though the patterns are generally identical, the variations from one census to the next give a more nuanced picture. Hence, for example, the improvement in Moroccan age reporting indicated by the original Whipple’s index is only partial, since only digit preference for ages ending in 0 and 5 is taken into account. Yet Myers’ blended index and the W_tot index, which also take account of digit preferences for other ages, show that the improvement in age reporting quality calculated in this way is much larger than that measured by the original Whipple’s index. In fact, for both countries, the original Whipple’s index underestimates the improvement in reporting quality.

33Last, Myers’ blended index and the total modified Whipple’s index, both calculated over the age range 23-62 years, vary in a practically identical manner, to within a few details, confirming the pertinence and validity of the W_tot index. The differences simply reflect the methods and assumptions upon which the two indices are based. The calculation method for the total modified Whipple’s index is simpler than that used for Myers’ blended index. Moreover, its comparison with the original Whipple’s index, the measure most widely used up to now, is robust, since both indices are based on identical principles and assumptions.

34In short, while the original Whipple’s index only measures preference for ages ending in digits 0 and 5, the modified total Whipple’s index (W_tot) takes account of preference and avoidance of all ten digits using all the information obtained via the specific W_i indices. Moreover, it produces practically the same results as Myers’ blended index. Its main advantage lies in the simplicity of its calculation method and its comparability with the original Whipple’s index. Hence, by taking account of the effects of all ten digits, the W_tot index provides an essential complement to the specific W_i indices and a more accurate measure of overall age reporting quality.

35Acknowledgements : The author wishes to thank Reto Schumacher and Philippe Wanner for kindly providing the Swiss census data, Aziz Ajbilou and Dominique Tabutin for supplying those of Morocco, and two anonymous reviewers for their helpful comments that led to improvements in the quality of this paper.